Bosonic qubits encoded in continuous-variable systems provide a promising alternative to two-level qubits for quantum computation and communication. So far, photon loss has been the dominant source of errors in bosonic qubits, but the significant reduction of photon loss in recent bosonic qubit experiments suggests that dephasing errors should also be considered. However, a detailed understanding of the combined photon loss and dephasing channel is lacking. Here, we show that, unlike its constituent parts, the combined loss-dephasing channel is non-degradable, pointing towards a richer structure of this channel. We provide bounds for the capacity of the loss-dephasing channel and use numerical optimization to find optimal single-mode codes for a wide range of error rates.
Bosonic qubits encoded in continuous-variable systems provide a promising alternative to two-level qubits for quantum computation and communication. So far, photon loss has been the dominant source of errors in bosonic qubits, but the significant reduction of photon loss in recent bosonic qubit experiments suggests that dephasing errors should also be considered. However, a detailed understanding of the combined photon loss and dephasing channel is lacking. Here, we show that, unlike its constituent parts, the combined loss-dephasing channel is non-degradable, pointing towards a richer structure of this channel. We provide bounds for the capacity of the loss-dephasing channel and use numerical optimization to find optimal single-mode codes for a wide range of error rates.
Quantum optical systems are typically affected by two types of noise: photon loss and dephasing. Despite extensive research on each noise process individually, a comprehensive understanding of their combined effect is still lacking. A crucial problem lies in determining the values of loss and dephasing for which the resulting loss-dephasing channel is antidegradable, implying the absence of codes capable of correcting its effect or, alternatively, capable of enabling quantum communication. A conjecture [Quantum 6, 821 (2022)] suggested that the bosonic loss-dephasing channel is not antidegradable if the loss is below 50. In this paper we refute this conjecture, specifically proving that for any value of the loss, if the dephasing is above a critical value, then the bosonic loss-dephasing channel is antidegradable. While our result identifies a large parameter region where quantum communication is not possible, we also prove that if two-way classical communication is available, then quantum communication---and thus quantum key distribution---is always achievable, even for high values of loss and dephasing.
Bosonic codes, leveraging infinite-dimensional Hilbert spaces for redundancy, offer great potential for encoding quantum information. However, a practical continuous-variable bosonic code that can simultaneously correct both photon loss and dephasing errors, while achieving a high level of compliance with the Knill-Laflamme conditions within an experimentally friendly structure, remains elusive. Here, we propose a code based on the superposition of squeezed Fock states with an error-correcting capability that scales as ∝exp(-7r), where r is the squeezing level. The codewords remain orthogonal at all squeezing levels. In particular, this code achieves high-precision error correction for both single-photon loss and dephasing, even at moderate squeezing levels. Building on this code, we develop quantum error correction schemes that exceed the breakeven point, supported by analytical derivations of all necessary quantum gates. Our code offers a competitive alternative to previous encodings for quantum computation using continuous bosonic qubits.
Quantum computation and communication are important branches of quantum information science. However, noise in realistic quantum devices fundamentally limits the utility of these quantum technologies. A conventional approach towards large-scale and fault-tolerant quantum information processing is to use multi-qubit quantum error correction (QEC), that is, to encode a logical quantum bit (or a logical qubit) redundantly over many physical qubits such that the redundancy can be used to detect errors. The required resource overhead associated with the use of conventional multi-qubit QEC schemes, however, is too high for these schemes to be realized at scale with currently available quantum devices. Recently, bosonic (or continuous-variable) quantum error correction has risen as a promising hardware-efficient alternative to multi-qubit QEC schemes. In this thesis, I provide an overview of bosonic QEC and present my contributions to the field. Specifically, I present the benchmark and optimization results of various single-mode bosonic codes against practically relevant excitation loss errors. I also demonstrate that fault-tolerant bosonic QEC is possible by concatenating a single-mode bosonic code with a multi-qubit error-correcting code. Moreover, I discuss the fundamental aspects of bosonic QEC using the framework of quantum communication theory. In particular, I present improved bounds on important communication-theoretic quantities such as the quantum capacity of bosonic Gaussian channels. Furthermore, I provide explicit bosonic error correction schemes that nearly achieve the fundamental performance limit set by the quantum capacity. I conclude the thesis with discussions on the importance of non-Gaussian resources for continuous-variable quantum information processing.
We present a systematic study of Schrödinger cat codes constructed from Kerr-type coherent states, including displaced Kerr coherent states and Barut--Girardello Kerr coherent states, each admitting two distinct families determined by the sign of the Kerr nonlinearity. By tuning the Kerr parameter and coherent-state amplitude, these states interpolate between su(2), su(1,1) coherent states, providing a unified and versatile foundation for this type of bosonic quantum error correction. Unlike standard two-component Schrödinger cat codes, where a single photon-loss event induces an uncorrectable bit-flip, the nonlinear phase-space structure of Kerr cat states enables simultaneous detection and correction of both photon-loss and dephasing errors within a unified recovery framework, with optimal recovery operations determined via convex optimization. We demonstrate that Kerr cat encodings significantly outperform conventional cat codes under combined loss and dephasing noise, and that judicious parameter optimization can suppress both error channels to a level that reduces the overhead of additional error correction layers. We further show that Kerr-deformed coherent-state manifolds under engineered two-photon driving emerge as effective steady states of driven-dissipative dynamics, with single-photon decoherence strongly suppressed and leakage outside the protected manifold appearing only as higher-order corrections in the deformation strength. Our extended formalism identifies generalized Kerr Schrödinger cat codes as promising candidates for fault-tolerant bosonic quantum computation in experimental platforms such as nonlinear photonics.
The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit in an oscillator has recently been followed by cat- and binomial-code proposals. Numerically optimized codes have also been proposed, and we introduce codes of this type here. These codes have yet to be compared using the same error model; we provide such a comparison by determining the entanglement fidelity of all codes with respect to the bosonic pure-loss channel (i.e., photon loss) after the optimal recovery operation. We then compare achievable communication rates of the combined encoding-error-recovery channel by calculating the channel's hashing bound for each code. Cat and binomial codes perform similarly, with binomial codes outperforming cat codes at small loss rates. Despite not being designed to protect against the pure-loss channel, GKP codes significantly outperform all other codes for most values of the loss rate. We show that the performance of GKP and some binomial codes increases monotonically with increasing average photon number of the codes. In order to corroborate our numerical evidence of the cat-binomial-GKP order of performance occurring at small loss rates, we analytically evaluate the quantum error-correction conditions of those codes. For GKP codes, we find an essential singularity in the entanglement fidelity in the limit of vanishing loss rate. In addition to comparing the codes, we draw parallels between binomial codes and discrete-variable systems. First, we characterize one- and two-mode binomial as well as multiqubit permutation-invariant codes in terms of spin-coherent states. Such a characterization allows us to introduce check operators and error-correction procedures for binomial codes. Second, we introduce a generalization of spin-coherent states, extending our characterization to qudit binomial codes and yielding a multiqudit code.
The pure-loss channel is a fundamental noise model for bosonic quantum platforms, characterised by a single parameter, the transmissivity. In realistic scenarios such as free-space quantum communication, the transmissivity fluctuates from one channel use to another, and the channel is a convex combination of pure-loss channels, known as a fading channel. Despite its practical relevance, its quantum Shannon theory has remained largely unexplored. Here we investigate degradability, anti-degradability, entanglement breakingness, and capacities of the fading channel. We prove that entanglement distribution and quantum key distribution can be achieved at a strictly positive rate over any fading channel that is not completely noisy. When the transmissivity takes a finite set of values, we determine the energy-unconstrained two-way quantum and secret-key capacities exactly, as the averages of those of the pure-loss components. We prove that thermal states, optimal for a broad class of static bosonic Gaussian channels, do not in general achieve the entanglement-assisted classical capacity of fading channels: for a binary fading model we derive the capacity-achieving state in closed form, and we exhibit channels for which non-Gaussian Fock-diagonal states strictly outperform every Gaussian encoding. For the quantum capacity, we give a simple sufficient condition on the transmissivity distribution under which weak thermal inputs yield a strictly positive rate. Outside this condition, we numerically identify parameter regions where no thermal input compatible with the energy constraint has a positive coherent information, while optimized non-Gaussian inputs do. For general fading distributions, we design an iterative variational algorithm to optimize the coherent and mutual information. Our work advances the study of quantum communication in the non-Gaussian regime.